Multiplicative Diophantine Approximation on Planar Lines with Restricted Denominators

We prove a Khintchine result for convergence of a multiplicative Diophantine set with restricted denominators on an arbitrary non-degenerate line. Specifically, given sequences of real numbers $\{a_n{\}}_{n\in \mathbb{N}},\{b_n{\}}_{n\in \mathbb{N}},\{c_n{\}}_{n\in \mathbb{N}},\{d_n{\}}_{n\in \mathbb{N}},$ we determine convergence conditions under which the set of $x\in [0,1]$ which satisfy $\parallel a_{n}x+c_n\parallel \cdot \parallel b_{n}x+d_n\parallel <\psi (n)$ for infinitely many $n\in \mathbb{N}$ has zero Hausdorff s-measure. We also obtain an upper bound for the Hausdorff dimension in the inhomogeneous setting.